Showing the chords of Nutcracker and Symphony no. 5

This graph shows the chordograms of Mahler’s Symphony no. 5 in C sharp minor: 4. Adagietto (sehr langsam) and Tchaikovksy’s Nutcracker Op. 71, Act 2: No. 13. Waltz of the flowers.

The dark blue parts represents the tonal center of the chords. In Mahler’s (no5) chordogram, the tonal center seems to be around F major or F minor, as both of these paths are most clearly blue. Eventhough the tital of the symphony might indicate that this piece is in C# minor, this part of the symphony is performed in F major. The algorithm seems to have some trouble clearly distinguishing F major from F minor, but otherwise does show the tonal center most clearly around the correct chord. Chordograms are not (yet) perfect representations of musical harmony and it is likely that the amount of different musical instruments that are present in this piece have caused noise in the algorithm, which makes it somewhat less clear.

This noise is even more visible in the chordogram from the Nutcracker. Most of this piece is supposed to be in D major, but this is not very clear from the graph as the darker blue parts look to be spread out over multiple chords. Although it could be said that the D major chord is the darkest blue after the first +/- 70 seconds. However, in these first 70 seconds the graph is the clearest. During this time at the beginning of the piece, two oboes and a harp are playing the song’s intro in A7. After this point, the horns join in and shortly afterwards the rest of the orchestra At this point the chordogram becomes more noisy and unclear as the algorithm is clearly not able to handle a representation of all these different instruments yet.

Showing the keys of Nutcracker and Symphony no. 5

The keygrams I created were supposed to show the prevalent key in which these two pieces, Symphony no. 5 in C sharp minor: 4. Adagietto (sehr langsam) and Tchaikovksy’s Nutcracker Op. 71, Act 2: No. 13. Waltz of the flowers.

However, as with the chordograms of these two pieces, there seems to be some noise that obstructs the clarity of these keygrams. This time, the noise is more visible in the no. 5 keygrams than in the Nutcracker keygram. The no. 5 keygram does not show clear keys throughout the entire piece, whit only a couple of brief moments of exception - such as the D flat major key around 250 seconds. In castrast to the blurry chordogram, the Nutcracker keygram shows a clear prevalence of the D major key throughout the piece. In fact, where the tonal center of the chordogram was most clear for the first +/- 70 seconds, it seems to be the inverse with the keygram. The introduction of the song seems to be more noisy with energy spread out over various keys, whereas the middle part is far more clear.

Looking at the underlying structures of our favorite tracks with a ceptogram

These graphs show cepstrograms for Swan Lake, Op.20, Act 2: No. 10 (the Swan theme) and Symphony no. 3 in D Minor: V..

It is clear that there is a high magnitude for the 3rd coefficient throughout both tracks. However, with these Mel-frequency cepstral coefficients it is not entirely clear what the third coefficient exactly represents. If one was to listen to the track whilst following along with the cepstogram, it seems as though the magnitude for co3 is the highest at the moments when the theme is played in high tones.

For Swan Lake, this is when the oboe that plays the melody of the track. Once the rest of the orchestra joins in, the magnitude diminishes somewhat. At this point, the music also speeds up. At the end, there is another spike of magnitude when the other instruments fall away and the oboe is again clearly audible. When the oboe dies down, there also is a spike in co2, when the lower toned wind instruments finish the theme. For No. 3, coefficient 3 is the highest when the woodwinds blow their highest tones during the melody around the 20th second mark and at the end around 230-250 seconds. At these moments, the woodwinds are most clearly audible. It therefore seems that coefficient 3 for both tracks is related to the high tones of these instruments.

In the ‘sections’ graphs, it is clear that for no. 3 these tones are present throughout the track and especially in the first and last parts. For Swan Lake, these high tones are mostly present in the first part of the track, and gradually diminish.

The structures of the Swan theme made visible with chroma and timbre

In order to look further into Swan Lake, I have created self-similarity matrices demonstrating its chroma and timbre features.

In the chroma-based self-similarity matrix, there are no parallel diagonal lines visible. This means that there are no exact repetitions in the track. Even though the theme of the track seems to repeats itself, this is done by various different instruments and with slightly different melodies wrought trough it. Therefore, we can see that there are no exact repetitions of the theme in the chromagram. Around the 75-95 into the song, the matrix shows sudden yellow vertical lines. This points to a sudden musical novelty in the song, and in this case it marks the moment when the string orchestra starts playing a melody that is different from the theme of the song.

The timbre-based self-similarity matrix shows a clear divide of the track in 3 main sections. The first section from 0-50 seconds represents the moment when the oboe is almost solely audible. Hereafter, there is a section from 50-75 seconds, which is when the other wind instruments join into the theme. During the final section hereafter, the string orchestra takes over the theme. Finally, the most striking aspect of this matrix is the sudden burst of yellow at the end of the song, around the 150 second mark. This is the moment when the lower toned wind instruments finish the theme, as this is the only moment in the track when we can solely hear these lower tones. In the matrix, we can also see these tones fading out a little bit.

The subjects of this research: Tchaikovsky and Mahler

For this assignment I will be conducting an exploratory analysis of the works of Pyotr Ilyich Tchaikovsky (1840-1893) to Gustav Mahler (1860-1911). My reasons for choosing to compare these two artists are mostly personal. I do not know much about classical music, as I mostly tend to listen to contemporary music. An exception to this rule is Tchaikovsky, whose musical pieces I have enjoyed listening to ever since my mother and grandmother took me to see Swan Lake when I was young. When I was working on my thesis in the summer of 2020, I started listening to classical music more often and found it quite enjoyable and relaxing. However, I still do not know much about this genre of music and I certainly do not feel like I can identify the composer of a musical piece on the radio just by listening to it, as my mother can. Therefore, I wanted to take this opportunity to dive more deeply into this genre of music in order to increase my knowledge on its different musical elements and the ways in these can be used by composers to create their own distinct styles. Besides Tchaikovksy, I also chose to analyze works by Mahler, as this is my mother’s favorite composer. It would be interesting to see in what ways the styles of our favorite composers differ. Furthermore, Tchaikovsky’s pieces are seen as works of the Romantic period, whereas Mahler is seen as a mix of the 19th century Austro-German music tradition and the modernism of the early 20th century. I selected the works that I will analyze by adding the Spotify public playlists “This is Tchaikovsky” (60 tracks) and “This is Mahler” (70 tracks) together. I chose these two playlists, as they were created by Spotify in order to represent work typical for the composers. They are performed by a variety of orcestra’s, which means that there coudl be some deviations based on the perfomers’ interpretations. I will pay special attention to mine and my mother’s favorite tracks of these composers. For me, these are Swan Lake, Op.20, Act 2: No. 10 and The Nutcracker, Op 71, Act 2: No. 13, which coincidentally are also the most popular Tchaikovsky tracks. For my mother, these are Symphony No. 5 in C-sharp Minor: IV and Symphony no. 3 in D Minor: V..

The overall differences between these two composers

Composer Danceability Energy Key Loudness Mode Valence Tempo
Mahler 0.2116857 0.08691571 5 -24.06131 1 0.1032543 99.32761
Tchaikovsky 0.2560017 0.11786833 2 -22.62868 1 0.1389133 104.08008 
$x
[1] "Valence"

$y
[1] "Energy"

$size
[1] "Danceability"

$title
[1] "Interactive plot comparing Tchaikovsky and Mahler"

attr(,"class")
[1] "labels"

Figure 2. Interactive plot comparing Tchaikovsky and Mahler

I first created a table of the means of the variables danceability, energy, loudness, valence and tempo of both composers, to which I also added the statistical mode’s for key and mode. Tchaikovsky seems to be higher overall in danceability, energy, valence and tempo, whereas Mahler’s pieces have been played more loudly. Furthermore, Tchaikovsky has most often composed pieces in Key=2, meaning D (also C-double sharp, E-double flat) . For Mahler, the key in which he most often composed is F (also E-sharp, G-double flat). Both composers have most often composed in major.

In the graph underneath, I plotted the tracks for Tchaikovsky and Mahler on their valence and energy. The size of the triangles indicates the measure of danceability, with larger triangles meaning more danceable tracks. The trend for both composers is shown in the lines in the graph. The line representing Tchaikovsky lies higher than the line for Mahler, and the former also reaches further with valence. However, this far reach of the line is based mostly on one point, Dance des petits cygnes in Swan Lake. and is therefore not necessarily representable for his whole body of work. I also plotted the means for both composers’ valence and energy on the graph. Here we can see more clearly that both have composed pieces that are generally low in valence and energy. Looking at the distribution of points, it seems that Mahler especially has the greatest portion of his work in this area, with Tchaikovsky having a somewhat larger amount of variation. Furthermore, there seems to be quite a large variation in danceability, as the triangles vary in sizes. The most danceable tracks also have the highest valence. However there doesn’t seem to be a further correlation between valence and danceability from looking at this graph alone, as there are also tracks with higher than average danceability for lower amounts of valence.